Collinear Fractals and Bandt's Conjecture

TL;DR

This paper studies the topological properties of a family of fractals called collinear fractals, explores the connectedness locus similar to the Mandelbrot set, and proves the generalized Bandt's conjecture for large n.

Contribution

It provides a detailed answer to an open question about collinear fractals, extends the covering property technique, and proves the generalized Bandt's conjecture for n ≥ 21.

Findings

The connectedness locus $\\mathcal{M}_n$ has a regular-closed structure.

For n ≥ 21, the entire non-real part of $\mathcal{M}_n$ is in the closure of its interior.

The paper confirms the generalized Bandt's conjecture for sufficiently large n.

Abstract

For a complex parameter $c$ outside the unit disk and an integer $n\ge2$, we examine the $n$-ary collinear fractal $E(c,n)$, defined as the attractor of the iterated function system $\{\mbox{$f_k \colon \mathbb{C} \longrightarrow \mathbb{C}$}\}_{k=1}^n$, where $f_k(z):=1+n-2k+c^{-1}z$. We investigate some topological features of the connectedness locus $\mathcal{M}_n$, similar to the Mandelbrot set, defined as the set of those $c$ for which $E(c,n)$ is connected. In particular, we provide a detailed answer to an open question posed by Calegari, Koch, and Walker in 2017. We also extend and refine the technique of the covering property by Solomyak and Xu to any $n\ge2$. We use it to show that a nontrivial portion of $\mathcal{M}_n$ is regular-closed. When $n\ge21$, we enhance this result by showing that, in fact, the whole $\mathcal{M}_n\setminus\mathbb{R}$ lies within the closure of its interior, thus proving that the generalized Bandt's conjecture is true.